Theorem 2.1 [1]}
Suppose \(n=k_2+3d+r\geqslant 10k_1+13k_2+3\) , where \(k_1, k_2, d, r\)  are positive integers and \(r\leqslant 2\) . Then \(ex(n, F(k_1, k_2; 2))=\mbox{\rm {max}$\lbrace e(G_1(k_2, 2)), e(G_2(k_1, k_2, 2))\rbrace $}\) . Furthermore, the extremal graph is \(G_1(k_2, 2)\)  when \(k_1=1\)  and \(G_2(k_1, k_2, 2)\)  when \(k_1>1\) . In particular, \(G_2(k_1, k_2, 2)\)  is also an extremal graph when \(k_1=1\)  and \(r=1\)  or \(r=2\) .

